Estimates for differential operators with constant coefficients in a half-space

Necessary and sufficient conditions (and also more explicit sufficient conditions) are obtained for the validity of the following estimates for differential operators with constant coefficients in the half-space $\mathbf R_+^n=\{(x,t):x\in\mathbf R^{n-1},\ t\geqslant0\}$:
\begin{gather*} \|\mathscr R(D)u\|^2\leqslant C\|\mathscr P(D)u\|^2,\qquad u\in C_0^\infty(\mathbf R_+^n),\quad (\mathscr Q_j(D)u)(x;0)=0\ (j=1,\dots,N), \\ \|\mathscr R(D)u\|^2\leqslant C\biggl(\|\mathscr P(D)u\|^2+\sum_{j=1}^N\langle\!\langle\mathscr Q_j(D)u\rangle\!\rangle _{s_j}^2\biggr), \end{gather*}
where ${\|\cdot\|}$ and $\langle\!\langle\,\cdot\,\rangle\!\rangle$ are the norms in $L_2(\mathbf R_+^n)$ and $H_s(\partial\mathbf R_+^n)$,
$$ D=\biggl(\frac1i\,\frac\partial{\partial x_1},\dots,\frac1i\,\frac\partial{\partial x_{n-1}};\frac1i\,\frac\partial{\partial t}\biggr), $$
and $C_0^\infty(\mathbf R_+^n)$ is the space of restrictions to $\mathbf R_+^n$ of functions in $C_0^\infty(\mathbf R^n)$.
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